Within the bonds of marriage, sexual intimacy is for the purpose of procreation (having children), the emotional enhancement between a husband and wife (being one flesh) and for their enjoyment. There are many reasons why two should not be sexually intimate outside of the bonds of marriage; one reason is the one flesh bonding.1
As to who is married and who isn't, God told early Christian Church leaders that whatever they bound on earth would be bound in heaven and whatever they loosed on earth would be loosed in heaven.2 I personally do not know of any Christian denomination which does not consider the marriage ceremony an important part of marriage (especially for Christians.) In fact, it is so vital, that without it, there is no marriage under normal circumstances.
The question might be asked, "Was there a ceremony when Adam and Eve were married?" The Bible doesn't say. It does say, however, that the Lord brought Eve to Adam to be Adam's wife.3 It is possible that Christ Jesus joined Adam and Eve together with some sort of ceremony. How is this possible? Well He appeared quite often in Old Testament days as the Angel of the Lord (before He became a man a little over 2000 years ago at Bethlehem, Judea.)
It is God who instituted government; that there might not be anarchy upon the earth.4 We are to obey the governing authorities. All governments, that I know of, have certain requirements that must be met in order for a marriage to be legal.
1 (Genesis 2:24; 1 Corinthians 6:16)
2 (Matthew 16:19, 18:18)
3 (Genesis 2:22)
4 (Romans 13:1-2)